#8158941

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(2,6)(3,9)(8,14)(10,11)(17,18,23,24)(19,20,21,25)(22,30,31,26)(27,32,29,28)(33,34,40,39)(35,43,38,41)(36,46,42,37)(44,45,47,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(18,24)(20,25)(26,30)(28,32)(33,35,40,38)(34,41,39,43)(36,47,42,44)(37,48,46,45);(3,9)(7,13)(10,11)(15,16)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28)(33,36)(34,37)(35,44)(38,47)(39,46)(40,42)(41,45)(43,48);(1,3,5,9)(2,7,6,13)(4,11,12,10)(8,16,14,15)(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29)(33,37,40,46)(34,42,39,36)(35,45,38,48)(41,47,43,44);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,21,23,19)(18,20,24,25)(22,29,31,27)(26,28,30,32)(33,38,40,35)(34,43,39,41)(36,47,42,44)(37,48,46,45);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,22,23,31)(18,26,24,30)(19,27,21,29)(20,28,25,32)(33,39,40,34)(35,43,38,41)(36,46,42,37)(44,48,47,45)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8066137);